The Einstein-Schrodinger theory without a cosmological constant was originally proposed by
Einstein
and Straus in 1946. In this form it is also known as
"Einstein's Unified Field Theory", or the
"Einstein-Straus Theory", or the
"Hermitian Theory of Relativity", or the
"Generalized Theory of Gravitation", or the
"Relativistic Theory of the Non-symmetric Field".
In 1947 Schrodinger
showed that the theory can be derived from a very
simple Lagrangian density if a cosmological constant is included.
This more general theory is usually called
"Schrodinger's Affine Field Theory" or the
"Einstein-Schrodinger Theory".
The Einstein-Schrodinger theory is a generalization of ordinary general relativity
which allows a non-symmetric fundamental tensor. When Einstein and
Schrodinger developed the theory they suspected that this fundamental tensor might
contain both the metric and electromagnetic field, and they hoped that the theory would
unify gravitation and electromagnetism. However, using the EIH method
it was shown way back in 1953 that the original theory apparently does not predict a
Lorentz force between charged particles.
Since then there has been some interest in the theory by a few researchers who dispute
the proof of the missing Lorentz force. Other researchers have assumed that the theory
is correct but represents something other than gravity and electromagnetism. Still others
have investigated theories which are significantly modified versions of the original
theory. In general though, there has not been much interest in the Einstein-Schrodinger
theory since the 1950s.
Here are a few of Einstein's and Schrodinger's original papers
on this topic.

Lambda-renormalized Einstein-Schrodinger theory:
Recently it has been shown that a well motivated modification of the
Einstein-Schrodinger theory does indeed closely approximate ordinary
general relativity and electromagnetism (Einstein-Maxwell theory),
the modification being the addition of a large cosmological constant.
This cosmological constant which multiplies the symmetric metric is assumed
to be nearly cancelled by Schrodinger's ``bare'' cosmological constant which
multiplies the nonsymmetric fundamental tensor, resulting in a total "physical"
cosmological constant which matches measurement. The fact that these two
cosmological constants multiply different fields has the effect of creating
a Lorentz force, and also fixes some other problems with the original theory.
One possible origin of the large cosmological constant is from zero-point
fluctuations. It is reasonable to assume that the Einstein-Schrodinger theory
must eventually be quantized to accurately predict reality, and this
cosmological constant can be viewed as a kind of zeroth order quantization effect.
The cancellation of cosmological constants resembles
renormalization
in quantum field theory, as in the cancellation of large particle ``self energies''
with large ``bare'' masses resulting in small ``physical'' masses.
Another possible origin of the large cosmological constant is from a Higgs field
vacuum expectation value as in GUT theories. The Lambda-renormalized Einstein-Schrodinger
theory closely approximates Einstein-Maxwell theory regardless of the origin of the
cosmological constant, and it becomes exactly Einstein-Maxwell theory in the limit
as this cosmological constant goes to infinity. In addition, when the fields are
generalized to be composed of Hermitian matrices, the theory closely approximates
Einstein-Maxwell-Yang-Mills theory. This suggests several possibilities for a complete
unified theory by using the Fermion and Higgs terms of GUT theories as sources.